Some numerical invariants of local rings

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Abstract

Let $R$ be a formal power series ring over a field of
characteristic
zero and $I\subseteq R$ be any ideal. The aim of this work is to
introduce some numerical invariants of the local rings $R/I$ by
using theory of algebraic $\mathcal D$-modules. More precisely, we will
prove that the multiplicities of the characteristic cycle of the
local cohomology modules $H_I^{n-i}(R)$ and
$H_{\mathfrak{p}}^p(H_I^{n-i}(R))$, where $\mathfrak{p} \subseteq R$ is
any prime
ideal that contains $I$, are invariants of $R/I$.